5.06 Trigonometric equations
Graphical solutions to trigonometric equations
The graphical solution to trigonometric equations requires that both sides of the equation be considered as separate functions. These functions are then plotted and the intersection points between the two functions are the solutions to the equation. The simpler equations can be solved in either degrees or radians if one of the functions is a constant value or both sides of the equation involve a trigonometric function. However, remember, if there is a trigonometric function and another type of function, the trigonometric function will need to be considered in radians to ensure equality of axis scales.
Worked examples
Example 1
Find all the solutions to the equation $\sin\left(x-60^\circ\right)=1$sin(x−60°)=1 over the domain of $(-360,360).$(−360,360).
Think: Graphically speaking, this is the same as finding the $x$x-values that correspond to the points of intersection of the curves $y=\sin\left(x-60^\circ\right)$y=sin(x−60°) and $y=1$y=1. As we are working with a constant value for one of the functions we can work in degrees for this problem.
Do: Plotting both graphs:
 |
| $y=\sin\left(x-60^\circ\right)$y=sin(x−60°) (green) and $y=1$y=1 (blue). |
We can see in the region given by $\left(-360^\circ,360^\circ\right)$(−360°,360°) that there are two points where the two functions meet.
 |
| Points indicating where the two functions meet. |
Since we are fortunate enough to have gridlines, the $x$x-values for these points of intersection can be easily deduced. Each grid line is separated by $30^\circ$30°, which means that the solution to the equation $\sin\left(x-60^\circ\right)=1$sin(x−60°)=1 in the region $\left(-360^\circ,360^\circ\right)$(−360°,360°) is given by:
$x=-210^\circ,150^\circ$x=−210°,150°
We can only solve equations graphically if the curves are drawn accurately and to scale. You won't be expected to solve equations graphically if it requires drawing the curves by hand.
Using technology
Example 2
Using technology, determine the solutions to $e^{x-4}=\sin(2x)$ex−4=sin(2x). over the interval $[0,360]$[0,360].
Think: You can use your graphics calculator or other technology to solve the above equation for you.
Do:
Using technology, we obtain the following values of $x$x:
$x=0.516^\circ,87.7^\circ,198^\circ,229^\circ$x=0.516°,87.7°,198°,229°
Practice questions
Question 1
Consider the function $y=2\sin2x$y=2sin2x.
Draw the function $y=2\sin2x$y=2sin2x.
Loading Graph...State the other function you would draw in order to solve the equation $2\sin2x=1$2sin2x=1 graphically.
Draw the line $y=1$y=1 below.
Loading Graph...Hence, state all solutions to the equation $2\sin2x=1$2sin2x=1 over the domain $\left[-180^\circ,180^\circ\right]$[−180°,180°]. Give your answers in degrees separated by commas.